Chapter 1.6, Problem 29P

This problem demonstrates the variation-of-parameters method for first-order linear differential equations. Consider the first-order linear differential equation.

$y^{\text{'}}+p\left(x\right)y=q\left(x\right)\left(\mathrm{1.6.15}\right)$

(a) Show that the general solution to the associated homogeneous equation

$y^{\text{'}}+p\left(x\right)y=0$

is

$y_H\left(x\right)=c_1{e}^{-{\displaystyle \int p\left(x\right)dx}}$

(b) Determine the function $u\left(x\right)$ such that

$y\left(x\right)=u\left(x\right)e^{-{\displaystyle \int p\left(x\right)dx}}$

is a solution to (1.6.15) and hence derive the general solution to (1.6.15)